1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
|
SUBROUTINE ZGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO )
*
* -- LAPACK routine (version 3.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2006
*
* .. Scalar Arguments ..
INTEGER IHI, ILO, INFO, LDA, N
* ..
* .. Array Arguments ..
COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* Purpose
* =======
*
* ZGEHD2 reduces a complex general matrix A to upper Hessenberg form H
* by a unitary similarity transformation: Q**H * A * Q = H .
*
* Arguments
* =========
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* ILO (input) INTEGER
* IHI (input) INTEGER
* It is assumed that A is already upper triangular in rows
* and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
* set by a previous call to ZGEBAL; otherwise they should be
* set to 1 and N respectively. See Further Details.
* 1 <= ILO <= IHI <= max(1,N).
*
* A (input/output) COMPLEX*16 array, dimension (LDA,N)
* On entry, the n by n general matrix to be reduced.
* On exit, the upper triangle and the first subdiagonal of A
* are overwritten with the upper Hessenberg matrix H, and the
* elements below the first subdiagonal, with the array TAU,
* represent the unitary matrix Q as a product of elementary
* reflectors. See Further Details.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* TAU (output) COMPLEX*16 array, dimension (N-1)
* The scalar factors of the elementary reflectors (see Further
* Details).
*
* WORK (workspace) COMPLEX*16 array, dimension (N)
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value.
*
* Further Details
* ===============
*
* The matrix Q is represented as a product of (ihi-ilo) elementary
* reflectors
*
* Q = H(ilo) H(ilo+1) . . . H(ihi-1).
*
* Each H(i) has the form
*
* H(i) = I - tau * v * v**H
*
* where tau is a complex scalar, and v is a complex vector with
* v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
* exit in A(i+2:ihi,i), and tau in TAU(i).
*
* The contents of A are illustrated by the following example, with
* n = 7, ilo = 2 and ihi = 6:
*
* on entry, on exit,
*
* ( a a a a a a a ) ( a a h h h h a )
* ( a a a a a a ) ( a h h h h a )
* ( a a a a a a ) ( h h h h h h )
* ( a a a a a a ) ( v2 h h h h h )
* ( a a a a a a ) ( v2 v3 h h h h )
* ( a a a a a a ) ( v2 v3 v4 h h h )
* ( a ) ( a )
*
* where a denotes an element of the original matrix A, h denotes a
* modified element of the upper Hessenberg matrix H, and vi denotes an
* element of the vector defining H(i).
*
* =====================================================================
*
* .. Parameters ..
COMPLEX*16 ONE
PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) )
* ..
* .. Local Scalars ..
INTEGER I
COMPLEX*16 ALPHA
* ..
* .. External Subroutines ..
EXTERNAL XERBLA, ZLARF, ZLARFG
* ..
* .. Intrinsic Functions ..
INTRINSIC DCONJG, MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
INFO = -2
ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZGEHD2', -INFO )
RETURN
END IF
*
DO 10 I = ILO, IHI - 1
*
* Compute elementary reflector H(i) to annihilate A(i+2:ihi,i)
*
ALPHA = A( I+1, I )
CALL ZLARFG( IHI-I, ALPHA, A( MIN( I+2, N ), I ), 1, TAU( I ) )
A( I+1, I ) = ONE
*
* Apply H(i) to A(1:ihi,i+1:ihi) from the right
*
CALL ZLARF( 'Right', IHI, IHI-I, A( I+1, I ), 1, TAU( I ),
$ A( 1, I+1 ), LDA, WORK )
*
* Apply H(i)**H to A(i+1:ihi,i+1:n) from the left
*
CALL ZLARF( 'Left', IHI-I, N-I, A( I+1, I ), 1,
$ DCONJG( TAU( I ) ), A( I+1, I+1 ), LDA, WORK )
*
A( I+1, I ) = ALPHA
10 CONTINUE
*
RETURN
*
* End of ZGEHD2
*
END
|
